Test 10

Pure Mathematics – Calculus

p.1

King’s College 2007 – 2008 F.6 Pure Mathematics Revision Test 10 Time Allowed: 60 minutes Total Mark: 30 1.

(2003)

x Using the substitution t = tan , evaluate 2

2.

(2004)

Evaluate (a)

3.

(2004)

Evaluate

4.

(2006)

Evaluate lim sin x sin

5.

(2006) (a) Let f : ℜ → ℜ and g : ℜ → ℜ be continuous on [a, b] and differentiable in (a, b), where a < b.

lim(tan 3 x + cos 4 x ) x , 1

x →0

3

(

(3 marks)

)

(b) lim cos 2004 + x − cos x . x →∞

x →0

∫ sec

dx

∫ 2 + cos x .

θdθ .

(6 marks)

(3 marks) 1 . x

(3 marks)

Suppose that g (a ) ≠ g (b ) and g ' ( x ) ≠ 0 for all x ∈ (a, b ) . Define h( x ) = f ( x ) − f (a ) −

f (b ) − f (a ) (g (x ) − g (a )) for all x ∈ ℜ . g (b ) − g (a )

(i) Find h(a) and h(b). (ii) Using Mean Value Theorem, prove that there exists β ∈ (a, b ) such that f ' (β ) f (b ) − f (a ) = . g ' (β ) g (b ) − g (a ) (5 marks) (b) Let u : ℜ → ℜ be twice differentiable. For each x ∈ ℜ , let F : ℜ → ℜ and G : ℜ → ℜ be defined by F (t ) = u ( x ) − u (t ) − u ' (t )(x − t ) and G (t ) =

γ ∈ I such that

( x − t )2 2

. For each c ≠ x , prove that there exists

F ' (γ ) F (c ) u ' ' (γ ) (x − c )2 , where I is the = and u ( x ) = u (c ) + u ' (c )( x − c ) + G ' (γ ) G (c ) 2

open interval with end points c and x. v( x ) = 2006 . x →0 x

(5 marks)

(c) Let v : ℜ → ℜ be twice differentiable. It is given that lim

(i) Prove that v(0) = 0. Hence find v'(0). (ii) Suppose that v' ' ( x ) ≥ 2 for all x ∈ ℜ . Prove that v( x ) ≥ 2006 x + x 2 for all x ∈ ℜ . (5 marks)

--End of Test--